PHYS 101: Lecture 3

Physics 101 Lecture 3Kinematics: Vectors and Motion in 1 Dimension

PHYS 101: Lecture 3

Review! Kinematics: Free Fall, A Special Case

Free Fall: motion caused by gravity alone

𝑎 = 𝑔, the acceleration of gravity

𝑔 = 9.8 m/s2

Important Kinematic Expressions

»𝑦 = 𝑦0 + 𝑣0𝑦𝑡 −1

2𝑔𝑡2

»𝑣𝑦 = 𝑣0𝑦 − 𝑔𝑡

»𝑣𝑦2 = 𝑣0𝑦

2 − 2𝑔 𝑦 − 𝑦0

PHYS 101: Lecture 3

Free Fall, in 2-dimensions

If you have free fall with both x and y motion, the motion in the x- and y-directions is independent

Free fall with horizontal motion

»𝑥 = 𝑥0 + 𝑣0𝑥𝑡»𝑣𝑥 = 𝑣0𝑥

»𝑦 = 𝑦0 + 𝑣0𝑦 𝑡 −1

2𝑔𝑡2

»𝑣𝑦 = 𝑣0𝑦 − 𝑔𝑡

»𝑣𝑦2 = 𝑣0𝑦

2 − 2𝑔 𝑦 − 𝑦0

No x acceleration!

Independence of x and y motion means they each do their own thing without the other caring about it

PHYS 101: Lecture 3

Tips for applying the kinematic equations

Always pick a point (a particular x and t, or y and t) in the trajectory and apply kinematic equation(s) for that point.

Commonly: Two unknowns after writing kinematic equations (e.g., maximum height reached by a ball and time it took to get there).

Don’t give up because the equations are hard. Use algebra to solve for unknowns. (Recall: you need 2 eqns to solve for 2 unknowns)

PHYS 101: Lecture 3

ExampleA ball is thrown straight up with initial velocity of 15 m/s. How far above the point of release is the ball when its velocity is -10 m/s?

𝑀𝑒𝑡ℎ𝑜𝑑 1 (𝑜𝑛𝑒 𝑠𝑡𝑒𝑝): 𝑈𝑠𝑒 𝑣𝑦2 = 𝑣𝑜𝑦

2 − 2𝑔(𝑦 − 𝑦𝑜) and

solve for y (yo is 0 if you pick the origin where you threw ball).

𝑀𝑒𝑡ℎ𝑜𝑑 2 𝑡𝑤𝑜 𝑠𝑡𝑒𝑝𝑠 :First use vy = voy – gt to find the time when vy = -10 m/s

Then use y = yo + voyt – (1/2)gt2 to find y at that time.

PHYS 101: Lecture 3

A ball is thrown straight up in the air and returns to its initial position.

For the period of time that the ball is in the air, which of the following statements is true?

A - Both average acceleration and average velocity are zero.B - Average acceleration is zero but average velocity is not zero.C - Average velocity is zero but average acceleration is not zero.D - Neither average acceleration nor average velocity are zero.

Clicker Q

PHYS 101: Lecture 3

Free Fall Demo: Two Dropped Objects

Which will hit the ground first, the feather or the coin?

A) coin B) Same C) Feather

y

x

PHYS 101: Lecture 3

Clicker Q

The speed of an object in free fall (neglect air resistance!)

A. Always increases.

B. Is constant.

C. Always decreases.

D. May increase or decrease or be constant.

E. May increase or decrease but is never constant.

PHYS 101: Lecture 3

Free Fall Clicker Qs Fred throws a ball 30 mph vertically upward.

Which of the following statements are true about the ball’s velocityand acceleration. (Let up be the positive direction)

On the way up?

On the way down?

A) vy < 0 B) vy = 0 C) vy > 0

A) ay < 0 B) ay = 0 C) ay > 0

A) vy < 0 B) vy = 0 C) vy > 0

A) ay < 0 B) ay = 0 C) ay > 0

PHYS 101: Lecture 3

More Fred Clicker QFred throws a ball 30 mph vertically upward and then catches it again at the same height he threw it from.

What is the speed of the ball just prior to catching it? (Neglect air resistance)

1) v < 30 mph 2) v = 30 mph 3) v > 30 mph

PHYS 101: Lecture 3

Fred’s Free Fall Calculations

Fred throws a ball 30 m/s vertically upward.

What is the maximum height the ball reaches?

𝑣𝑦2 = 𝑣𝑜𝑦

2− 2𝑔Δ𝑦

Δ𝑦 = (𝑣𝑦2 − 𝑣𝑜𝑦

2 )/(−2𝑔)

=−(30)2

𝑚2

𝑠2

−2× 9.8m

s2

= 46 m.

What is the speed at max height?

Or, use: vtop,y = voy – gt to get t

Then y = yo + voyt – (1/2)gt2

𝑣𝑦 = 0𝑚/𝑠 at maximum height.

PHYS 101: Lecture 3

Fred’s Free Fall Calculations

Fred throws a ball 30 m/s vertically upward.

How long does it take to reach its maximum height?

𝑣𝑦 = 𝑣0𝑦 − 𝑔𝑡

𝑡 =𝑣𝑦−𝑣

0𝑦

𝑎

=0 – 30

m

s

−9.8m

s2

= 3.1 seconds

How long does it take the ball to make the complete round trip?

PHYS 101: Lecture 3

Dennis and Carmen are standing on the edge of a cliff.

Dennis throws a basketball vertically upward.

At the same time Carmen throws a basketball vertically downward.

Both balls are thrown with the same initial speed.

Whose ball hits the ground first?

A. Dennis' ball

B. Carmen's ball

C. Same

v0

v0

DennisCarmen

H

vA vB

More free fall clicker Qs

𝑦 = 𝑦𝑜 + 𝑣𝑜𝑦𝑡 + ½𝑎𝑦𝑡2

Dennis: 0 = 𝐻 + 𝑣𝑜𝑦𝑡 − ½𝑔𝑡2

Carmen: 0 = 𝐻 − 𝑣𝑜𝑦𝑡 − ½𝑔𝑡2

PHYS 101: Lecture 3

Dennis and Carmen are standing on the edge of a cliff.

Dennis throws a basketball vertically upward.

At the same time Carmen throws a basketball vertically downward.

Both balls are thrown with the same initial speed.

Whose ball is moving fastest when it hits the ground?

A. Dennis' ball

B. Carmen's ball

C. Same

v0

v0

DennisCarmen

H

vA vB

More free fall clicker Qs

PHYS 101: Lecture 3

Free Fall Facts:

Remember: 𝑎𝑦 = −𝑔.

Velocity becomes more and more negative if object is dropped from rest.

If 𝑣𝑦 > 0, speed decreases.

If 𝑣𝑦 <= 0 speed increases.

PHYS 101: Lecture 3

Vectors

Vector:

A quantity that has:

size (called magnitude)

direction

In physics lots of quantities are vectors such as:

velocity

acceleration

forces

PHYS 101: Lecture 3

Vectors We represent vectors with arrows.

The length of the arrow is the magnitude of the vector

The direction is shown by which way the arrow points

Rules for dealing with vectors:

You can move a vector in space and it is the same vector, as long as you do not change:

» the magnitude (length of the arrow)

» the direction

These two red vectors are the same.

A Vector

PHYS 101: Lecture 3

Adding and subtracting vectors

To add two vectors, put the tail of the second vector at the tip of the first vector, and draw the sum from the tail of the first vector to the tip of the second.

Let’s perform Ԧ𝐴 + 𝐵 = Ԧ𝐶:

Ԧ𝐴 𝐵

Ԧ𝐴 𝐵+ =

Ԧ𝐴 𝐵

Ԧ𝐶

PHYS 101: Lecture 3

Subtracting Vectors

To subtract vectors:

add the negative of the vector you are subtracting.

The negative of a vector has the same length but points in the opposite direction.

─ Bറ𝐴 റ𝐶

=

PHYS 101: Lecture 3

Swimmer Example: Velocity is a Vector

Ann needs to swim across a river.

Ann can swim 5 mph with respect to the water.

The river flows 3 mph East (i.e. from Ann’s left to Ann’s right).

At what angle should Ann swim to end directly to the other side of the river?

Clicker Q: Step 1: In which direction should Ann swim?

a) toward her left

b) toward her right

c) straight across

x

y

River

3 mph

Ann: can swim 5 mph

Ann’s Destination

PHYS 101: Lecture 3

Swimmer Example: Velocity is a Vector

Ann needs to swim across a river.

Ann can swim 5 mph with respect to the water.

The river flows 3 mph East (i.e. from Ann’s left to Ann’s right).

At what angle should Ann swim to end directly to the other side of the river?

Step 2: The vector diagram for Ann’s swim will look like this:

x

y

River

3 mph

Ann: can swim 5 mph

Ann’s Destination

q

റ𝑣𝐴𝑛𝑛 റ𝑣𝑁𝑒𝑡

റ𝑣𝑤𝑎𝑡𝑒𝑟

q

റ𝑣𝐴𝑛𝑛 റ𝑣𝑁𝑒𝑡

റ𝑣𝑤𝑎𝑡𝑒𝑟

PHYS 101: Lecture 3

Swimmer Example: Velocity is a Vector

Ann needs to swim across a river.

Ann can swim 5 mph with respect to the water.

The river flows 3 mph East (i.e. from Ann’s left to Ann’s right).

At what angle should Ann swim to end directly to the other side of the river?

Step 3: Find 𝜃:

From the triangle, we know: sinq = vwater/vAnn = 3/5

So solve for q: q = arcsin (3/5) = 36.9o

x

y

q

റ𝑣𝐴𝑛𝑛 റ𝑣𝑁𝑒𝑡

റ𝑣𝑤𝑎𝑡𝑒𝑟

How would you go about finding the green vector, which is Ann’s velocity relative to the ground?

PHYS 101: Lecture 3

Checkpoint 3, Q3An object experiences a constant acceleration and attains a

velocity as shown in the figure below. Which of the following

vectors best corresponds to the initial velocity of the object?

PHYS 101: Lecture 3

If these two vectors are added together, which

drawing represents the resultant vector?

Checkpoint 3, Q1